Answered

Consider the graph of the line $y=\frac{1}{2} x-4$ and the point ($-4, 2$).



The slope of a line parallel to the given line is [__].



A point on the line parallel to the given line, passing through ($-4, 2$), is [__].



The slope of a line perpendicular to the given line is [__].



A point on the line perpendicular to the given line, passing through ($-4, 2$), is [__].

Verified Answer :

  • Determine the slope of the original line as \(m = \frac{1}{2}\).
  • Calculate the perpendicular slope as \(m_{\perp} = -2\).
  • Derive the equation of the perpendicular line using the point \((-4, 2)\) as \(y = -2x - 6\).
  • Select a point on the line, such as \((0, -6)\), which satisfies the equation.
  • Final answer: \(\boxed{(0, -6)}\).

Explanation

  1. Analyze the Given Information and Perpendicular Condition

We are given the line \(y = \frac{1}{2}x - 4\) and a point \(P = (-4, 2)\). Our objective is to find the equation of the line perpendicular to the given line that passes through point \(P\), and then identify a point on this perpendicular line. The slope of the given line \(y = \frac{1}{2}x - 4\) is \(m = \frac{1}{2}\). Two lines are perpendicular if the product of their slopes is \(-1\). Thus, if \(m_{\perp}\) is the slope of our perpendicular line, we must have \(m \cdot m_{\perp} = -1\).

  1. Find the Equation of the Perpendicular Line

Using the condition \(m \cdot m_{\perp} = -1\), where \(m = \frac{1}{2}\), we find the slope of the perpendicular line:
\[\frac{1}{2} \cdot m_{\perp} = -1 \implies m_{\perp} = -2.\]
Now, we use the point-slope form of a linear equation, \(y - y_1 = m_{\perp}(x - x_1)\), with our point \((-4, 2)\) and slope \(m_{\perp} = -2\):
\[y - 2 = -2(x - (-4))\]
\[y - 2 = -2(x + 4)\]
\[y - 2 = -2x - 8\]
\[y = -2x - 6\]
This is the equation of the line perpendicular to \(y = \frac{1}{2}x - 4\) passing through \((-4, 2)\).

  1. Identify a Point on the Perpendicular Line

To find a point on the line \(y = -2x - 6\), we can substitute any value for \(x\) and calculate the corresponding \(y\). Let's choose \(x = 0\):
\[y = -2(0) - 6 = -6.\]
Thus, the point \((0, -6)\) lies on the line. Alternatively, we could choose \(x = -3\):
\[y = -2(-3) - 6 = 6 - 6 = 0.\]
So, the point \((-3, 0)\) also lies on the line. We will state the point \((0, -6)\) as our answer.

Examples

In urban planning, architects often design roads or utility lines that must intersect at right angles (perpendicularly). If a main pipe follows a path defined by the function \(y = \frac{1}{2}x - 4\), a service pipe branching off at a specific location, such as \((-4, 2)\), must follow the perpendicular path \(y = -2x - 6\) to ensure proper drainage or flow connection. Finding a point on this new line helps planners mark exact excavation spots on a map.